Convex risk measures on spaces of c\`adl\`ag processes whose supremum
belongs to an Orlicz heart are discussed. We obtain representations of
linear functionals on such spaces, and introduce some examples of convex
risk measures. In addition, we deal with convex risk measures
associated to hedging and pricing for American options. Among others, we
look into shortfall risk measure in detail. We also give some remarks
on convex risk measures treated in this talk.